The

Context. Nearly 50% of all sources detected by the Fermi Large Area Telescope are classified as blazars or blazar candidates, one of the most elusive classes of active galaxies. Additional blazars can also be hidden within the sample of unidentified or unassociated γ-ray sources (UGSs) that constitute about one-third of all gamma-ray sources detected to date. We recently confirmed that the large majority of Fermi blazars of the BL Lac subclass have an X-ray counterpart. Aims. Using the X-ray properties of a BL Lac training set and combining these with archival multifrequency information, we aim to search for UGSs that could have a BL Lac source within their γ-ray positional uncertainty regions. Methods. We reduced and analyzed the Swift X-ray observations of a selected sample of 327 UGSs. We then compared the X-ray fluxes and hardness ratios of all sources detected in the pointed fields with those of known Fermi BL Lacs. Results. We find at least one X-ray source, lying within the γ-ray positional uncertainty at 95% confidence level, for 223 UGSs and a total of 464 X-ray sources in all fields analyzed. The X-ray properties of a large fraction of them, eventually combined with radio, infrared, and optical information, exhibit BL Lac multi-frequency behavior, thus allowing us to select high-confidence BL Lac candidates; some of them were recently observed during our optical spectroscopic campaign which confirmed their nature. Conclusions. We find that out of 50 X-ray sources that were confirmed as BL Lacs through optical spectroscopy, 12 do not show canonical mid-infrared or radio BL Lac properties. This indicates that the selection of X-ray BL Lac candidates is a strong method to find new counterparts within Fermi UGSs. Finally, we pinpoint a sample of 32 Swift/XRT candidate counterparts to Fermi UGSs that are most likely BL Lac objects